5.2 Greeks w.r.t the correlation in a stochastic volatility model
5.2.1 The independent case
We consider the follwing system
dSt=rtStdt+Stσ(Vt)dWt1 dVt=u(Vt)dt+v(Vt)dWt2 drt=f(rt)dt+g(rt)dWt3.
(5.83)
which can be represented in a matrix form as
d
St
Vt rt
=
rtSt
u(Vt) f(rt)
dt+
Stσ(Vt) 0 0 0 v(Vt) 0
0 g(rt)
d
Wt1 Wt2 Wt3
(5.84)
where {Wti : 0 ≤ t ≤ T} for i = 1,2,3 are uncorrelated independent Brownian motions. If we let
Xt =
St Vt rt
, A(Xt) =
rtSt u(Vt) f(rt)
,
5.2. GREEKS W.R.T THE CORRELATION IN A STOCHASTIC VOLATILITY MODEL75
and
B(Xt) =
Stσ(Vt) 0 0 0 v(Vt) 0
0 0 g(rt)
(5.85)
and
x=
S0 V0 r0
, Wt =
Wt1 Wt2 Wt3
where the process {Wt: 0≤t≤T} is a standard three-dimensional Brownian motion, then we obtain a stochastic differential equation given by
dXt=A(Xt)dt+B(Xt)dWt, X0 =x∈R. (5.86) We assume that the drift coefficientA(·) and the diffusion coefficientB(·) are bounded with partial derivatives and satisfy linear growth and the Lipschitz conditions. From the general formula from Theorem 5.0.3, since in this case our matrix B(Xt) from (5.85) is a strictly diagonal matrix, its inverse is given by
B−1(Xt) =
1
Stσ(Vt) 0 0 0 v(V1
t) 0
0 0 g(r1
t)
(5.87)
The first variation process Yt for Delta is given by
Yt=
St
S0
0 0
Thus
(B−1(Xt)Yt)T = YtT(B−1(Xt))T
=
St
S0 0 0
1
Stσ(Vt) 0 0 0 v(V1
t) 0
0 0 g(r1
t)
=
1
S0σ(Vt) 0 0
. (5.88)
where T denote the transpose. If we multiply (5.88) by a 3-dimensional Browinan motion column matrix we obtain
(B−1(Xt)Yt)TdWt =
1
S0σ(Vt) 0 0
dWt1 dWt2 dWt3
.
= 1
S0σ(Vt)dWt1. (5.89)
If we let the deterministic function a(t) = T1. Then the Malliavin weight function for Delta (∆) is given by
π∆= Z T
0
a(t)(B−1(Xt)Yt)TdWt= 1 T S0
Z T 0
1
σ(Vt)dWt1. For the computation of Vega, the first variational process Yt is given by
Yt =Yt=
St
S0
Vt
V0
0
. (5.90)
Similarly to Delta, we have
(B−1(Xt)Yt)TdWt = YtT(B−1(Xt))TdWt
=
St
S0
Vt
V0 0
1
Stσ(Vt) 0 0 0 v(V1
t) 0
0 0 g(r1
t)
dWt1 dWt2 dWt3
=
1 S0σ(Vt)
Vt
V0v(Vt) 0
dWt1 dWt2 dWt3
= 1
S0σ(Vt)dWt1+ Vt
V0v(Vt)dWt2. (5.91) If we again let a deterministic functiona(t) = T1, then the Maliavin weight function for Vega (V) is given by
πV = Z T
0
a(t)(B−1(Xt)Yt)TdWt= 1 T S0
Z T 0
1
σ(Vt)dWt1+ 1 T V0
Z T 0
Vt
v(Vt)dWt2. Lastly for the computation of Rho, the first variational process Yt is given by
Yt= ∂
∂r0St =
St
S0
0
rt
r0
(5.92)
5.2. GREEKS W.R.T THE CORRELATION IN A STOCHASTIC VOLATILITY MODEL77
. In the very same way, we have
(B−1(Xt)Yt)TdWt = YtT(B−1(Xt))TdWt
=
St
S0 0 rrt
0
1
Stσ(Vt) 0 0 0 v(V1
t) 0
0 0 g(r1
t)
dWt1 dWt2 dWt3
=
1
S0σ(Vt) 0 r rt
0g(rt)
dWt1 dWt2 dWt3
= 1
S0σ(Vt)dWt1+ rt
r0g(rt)dWt3. (5.93) If we again let a deterministic functiona(t) = T1, then the Maliavin weight function for Rho (ρ) is given by
πρ= Z T
0
a(t)(B−1(Xt)Yt)TdWt = 1 T S0
Z T 0
1
σ(Vt)dWt1+ 1 T r0
Z T 0
rt
g(rt)dWt3.
We note that in all the cases where we computed the price sensitivities, there are no direct computations of the derivative of the payoff functions. All the results we got indicates that the efficiency of the Malliavin calculus in the computation of price sensitivities does not depend on the type/nature of the payoff function. By making use of the integration by parts formula, we have indeed seen that a Greek can be represented as the expectation of the product of the payoff function and the Malliavin weight function.
Conclusion
This study was based on the computation of the hedging portfolios and the price sensitivities, known as Greeks, in the case where we have the discontinuous payoff functions using the Malliavin calculus approach. We introduced the importance and the background of Malliavin calculus and also what have been done before. We developed the Wiener’s construction of Brownian motion and the stochastic integral. We discussed some important properties of Malliavin calculus which includes essential tools such as the integration by parts formula.
This formula avoid the direct derivation of the functional, instead result in the product of the functional and the so called Malliavin weight function. The integration by parts formula plays a huge role in the computation of the price sensitivities. We only restricted ourselves to one dimensional case. The Clark-Ocone formula is used for the computation of hedging portfolios. We showed the Malliavin derivative of stochastic differential equation where the focus is on the diffusion process. As a result, we constructed the first variational process which is the partial derivative of the stochastic differential equation with respect to the initial condition.
For the application of the Malliavin calculus to mathematical finance, we used the Clark- Ocone formula to obtain the general representation formula of the replicating strategy. The general formula was applied to different types of payoff functions of the European type where we realised that the hedging portfolio is naturally related to the Malliavin derivative of the terminal payoff. In addition, we computed the price sensitivities in the Malliavin sense.
The Malliavin calculus properties are used to compute the general representation formula of price sensitivities which include the Delta (∆), Gamma (Γ) and Vega (V). We considered the geometric Brownian motion case as an example.
Further we computed the price sensitivities with respect to the correlation in a stochastic 78
79 volatility model. We considered the 2-dimensional correlated Brownian motion where we computed the sensitivity of the option with respect toρ. We considered also a 3-dimensional Brownian motion and compute the price sensitivity which includes the Delta, Vega and Rho.
We conclude our study by considering a 3-dimensional uncorrelated Brownian motion and compute again the Delta, Vega and Rho of the option price. We hope to extend the diffusion case to processes that include jumps. We would also like to apply similar concepts to option price of American type where the exercise of the option take place on or before the maturity time T.
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